L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s − 0.999·20-s − 0.999·22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + 13-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s − 0.999·20-s − 0.999·22-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9669901342\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9669901342\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.252663412021903219924910686469, −8.767698555582445620346881678275, −7.935990040521423602192507749582, −6.78590375526552868856074430115, −6.29535820623200154252166405697, −5.23755517648558563602637254330, −3.78399624958092782872798329789, −3.34759132323626274505927467079, −2.37606116979054780804943363142, −1.04343775571270355610714731420,
1.24734797279087097035514517596, 2.26930934259095252146458172564, 4.13755638718145392324490061925, 4.73052419325288315271235694660, 5.75578903014232352128065962670, 6.15014413120255621455739690476, 7.24760463368105756208923115384, 8.025278588947066673591393044354, 8.729595624659436134073416463658, 9.166821327070817930021156558081